Validation of an elastoplastic model for chalk

Computers and Geotechnics 9 (1990) 257-272

VALIDATION OF AN ELASTOPLASTIC MODEL FOR C H A L K

J.F. Shao, J.P. Henry Laboratory of Mechanics of Lille E.U.D.I.L. 59655 Villeneuve d'Ascq cedex,France

ABSTRACT A new elastoplastic model for porous rocks ( chalk ) has been developed [1]. It is based on the concept of two plastic deformation mechanisms [2]. In this paper, we present the validation of this model for both homogeneous tests and a typical boundary condition problem, hollow cylinder t e s t . Comparisons between experimental results and predictions by the model are presented for each test. Influences of parameter perturbations on numerical simulation are also studied.

INTRODUCTION Chalk is certainly one among the rock materials with the most complex rheological behaviour. In the recent years, a rheological understanding of this material has become essential in petroleum engineering. Indeed, as a consequence of the development of chalky oil deposits in the North Sea, very large compactions of hydrocarbon layers have been observed, and their transmission to the surface (subsidence) is considerable [3]. Numerous experimental studies have been carried out in the recent years for this rock [4] [5] [6]. The obtained results have shown that its behaviour is highly complex and exhibits very important plastic deformations due to the gradual destruction (collapse and distorsion ) of the porous structure, both under hydrostatic and deviatoric loadings. The high porosity of this material,which frequently exceeds 40% and can sometimes reach 45% or more, has a decisive impact on its mechanical behaviour. Under hydrostatic loading, the behaviour of chalk is very different from that of most Mohr-Coulomb rock materials. In figure 1,the typical results of a hydrostatic compression test for chalk are presented [4]. It can be dearly noticed that under increasing hydrostatic stress, the stress-strain curve exhibits three very distinct zones. Firstly, a linear elastic zone limited to a certain 257 Computers and Geotechnics 0266- 352X/90/S03.50 © 1 990 Flsevier Science Publishers Ltd, England. Printed in Greet Britain

258

hydrostatic effective pressure of about 10 MPa is observed. From this limit, a very important irreversible compaction appears with a great plastic strain rate and a smooth strain hardening. This zone covers a range of substantial strains. Lastly, the curve becomes inflected with an increasing strain hardening rate. Furthermore, the bulk modulus increases with loading [4]. Under deviatoric loading (triaxial compression tests), the behaviour of chalk is highly influenced by the hydrostatic stress (confining pressure). At lower confining pressure (< 5MPa), a linear elastic zone is observed during deviatoric loading and a pseudo-brittle peak for the stress strain curve can be obtained. When confining pressure increases (5MPa < Pc < 10MPa ), the linear elastic zone becomes smaller and no peak is observed. Lastly at higher confining pressure (> IOMPa), the elastic zone has completely disappeared and no failure of sample is observed until the axial strain exceeds about 12%, the hardening phase only is observed.

60 50 40 30 20

10 ,

I

4

i

I

8

i

|

i

12 16 Ev (%)

Figure 1: hydrostatic compression test for chalk From the above experimental observations [4], chalk offers essentially two distinct types of plastic yield mechanisms. An elastoplastic model has been developed to describe chalk behaviour. It is based on an elastoplastic model with two yield surfaces initially developed by Lade [2] for eohesionless soils. In the framework of applications of the proposed model in petroleum engineering, we have chosen some particular stress paths to test the validity of this model. For homogeneous case, we have used reduced triaxial compression test, proportional loading test, oeclometric test and undrained triaxial compression test. The validity of the model for a typical boundary condition problem, hollow cylinder test, is then examined. Influences of perturbations of the model parameters on the numerical predictions are also studied.

259 PRESENTATION OF THE ELASTOPLASTIC MODEL The elastoplastic model for chalk is developed starting from that proposed by Lade [2] for cohesionless sand. The model is formulated using the concept of two plastic deformation mechanisms [7] and the three dimensional failure criterion of Lade [9]. The total swain increment is divided into an elastic part d~j e, a collapse plastic part d~j c to interpret the plastic pore collapse of chalk, and a deviatoric part d~j d to interpret the plastic distorsion of chalk. We then have : d

(1)

deij = de~j + dE~j + dEij

Elastic strain The elastic strain increment is calculated by Hooke's law. The bulk modulus depends on the first invariant of stress tensor I t, and Poisson's ratio v is kept constant. The elastic modulus E is then determined from K and v. This is expressed by the relations (2) and (3) :

(1,7

(21

K = Ko ~,3po)

(3)

E=3(1-2v)

K

where Ko is the initial bulk modulus of chalk in the elastic domain and Po the hydrostatic initial elastic limit (figure 1). Tensile strength of chalk In order to take into account the tensile strength which can be sustained by chalk, a transformation of stress coordinates is performed according to Desai et al [8] and Lade[2]:

(4)

aij = aij + R PI &tj

where R is the tensile strength parameter, Pa the atmospheric pressure expressed in the same units as stresses, and 5ij is Kroneeker delta. The yield functions and plastic potentials in this model will thus be formulated in the transformed stress space.

260

Collapse plastic strain The collapse plastic strain describes the pore collapse of chalk. It is determined by an associated flow rule and an isotropic strain hardening law. In the transformed stress space, we have : collapse yied function, (5)

Fc= It - Yc = 0

where (6)

FI = o t + o 2 + 0 3

collapse plastic potential, (7)

Qc = F¢

collapse strain hardening law, (8)

Yc = yO + a . Pa" ~ " e×p (c.~¢)

and (9)

yO = 3(Po + R . P a )

where a, n and c are three material parameters, and ~ the equivalent collapse plastic strain [ 1 ]. Applying the theory of plasticity, we have :

(10)

dec. = d~. ~ Ij

where dk c is the collapse hardening coefficient. We notice that the collapse plastic strain is fully volumetric, deCij is then a spheric tensor and equal to (d~Ckk / 3)8ij.

Deviatoric plastic strain The deviatoric plastic strain describes the plastic distorsion of chalk. It is determined by a non-associated flow rule and an isotropic strain hardening law. Lade [9] proposed a three dimensional failure criterion for rock materials. Its validity was tested with experimental results for chalk [4]. A very good agreement between theoretical predictions and experimental data was obtained, so Lade's criterion has been chosen to define the failure surface for chalk. Then we

261 assumed that the deviatoric yield surface has the same general shape as the failure surface. In the transformed stress space, we have : failure criterion, m

iI

il

(11)

-

27

, -

Yd = 0

where m and ydr are dimensionless material paran~ters, and (12)

I3 = ffl " if2 " if3

deviatoric yield function, m

(13)

Fd =

- 27

~

- Yd = 0

deviatoric plastic potential, 3

(14)

Qd = i l -

27 i 3

deviatoricstrainhardeninglaw, -

(15)

o

yd=Yd +

b+~7

and o

(16)

Yd

=

t

~ a (Po - %)

o

,

Yd > 0

where t is a parameter to define the initial elastic limit of material, a o the initial hydrostatic stress, and ~d the equivalent deviatoric plastic strain. T is a dimensionless material parameter depending on the initial consolidation state of the material, b is a function of hydrostatic pressure to describe the pressure dependency of deviatoric strain hardening for chalk. The following expression is proposed for this dependence:

262

(17)

b=ln

/ / c~+~

b > 0

where ~ and [3 are two material parameters and

(18)

I 1 = 0-1 + 0"2+ 0"3

Using the plasticity theory, the deviatoric plastic strain is equal to : d

(19)

deij = d~.d - 20"..

U

where dk a is the deviatoric hardening coefficient.

VALIDATION OF THE CONSTITUTIVE MODEL FOR HOMOGENEOUS TESTS Two very pure white chalks extracted from two different quarries are used for the validation of the model. Their average porosity is about 45% and CaCo 3 content 96% [4]. The typical values of the model parameters for such very porous chalks are given in table 1. Table 1: Typical values of parameters for porous chalk V 0.15 0.18 m Ko

1400MPa

f Yd

59

Po

10 MPa

ct

0.98

R

7

13

0.00053

1800

T

0.95

n

0.41

t

0

c

8.50

Simulation of proportional loading test It consists of a special triaxial test in which two principal stresses 0-t and o 3 are increased simultaneously with a constant ratio k. The constitutive model is then used to simulate

263 this test. In figures 2 and 3, we present experimental results and numerical simulations by the model for two values of k . From these comparisons, it can be seen that there is a very good agreement for k -- 2.35, and for k -- 5.1 a satisfactory accuracy for the axial strain but some differences for volumetric strain.The model seems to overestimate the volumetric strain when the hydrostatic stress is small (when k is large ).

40 k=2.35 30 I,-4

r/3

20 10

Vffi

0

n

0 10

5

0

5

Ev (%)

10 E1 (%)

Figure 2: simulation of proportional test (k=2.35) 12

k=5.1

10

8 6

experiment

simulation

4 2

0 10 $ Ev (%)

0

5

10 E1 (%)

Figure 3 : simulation of proportional test (k=5.1)

Simulation of reduced triaxial compression test This is also a special triaxial test. Starting from a hydrostatic stress state, the confining stress o 3 is decreased, and the axial and volumetric strains are registed as a function of stress deviator. In figures (4a) and (4b), we present the experimental results and the numerical predictions given by the model. One can notice that there is a very good agreement for the axial strain ,but a certain difference is observed for the volumetric strain.

264 20

~

ox~~

15

f

10

simulation

5 a

0

I

=

I

10

0

n

20

El(E-3)

30

Figure 4a: simulation of reduced compression test -axial strain 20 15 10

0

-3

-2

-1

0

1

2 3 Ev (%)

Figure 4b: simulation of reduced compression test -volumetric strain Simulation of oedometric test

In this test, the lateral strain E3 is kept to zero, the axial strain 81 is increased. The axial stress •1 and the confining pressure 0 3 are measured as a function of E 1. In figures (5a) and (5b), comparisons between the experiment and the prediction are presented. Good agreement can be observed for this test.

50t On.'/ I 6O

40

simulati

20

' ; ~0

riment

. . . . .4

2

6

;

,'o

12

E1(%)

Figure 5a: simulation of oedometric test -axial stress

40 j 265

2O lO

0

- ~ T

•

0

simulation !

i

i

t

2

4

6

8

i

10 12 E1(%)

Figure 5b: simulation of oedometric test -conf'ming pressure

Simulation of undrained triaxial compression test Two undrained triaxial compression tests with different confining pressures are carried out. In the present study, the concept of Terzaghi effective stress is used in the numerical simulation. Then it is assumed that the volumetric strain of chalk is equal to zero under undrained condition. As the total confining pressure G 3 in conventional ~axial test is constant, the pore pressure Pi can be determined from the variation of the effective confining pressure dG 3' and we have dP i = - dG3'. In figures (6) and (7), stress deviator and pore pressure are expressed as function of axial strain. Comparisons between the experiment and the simulation show satisfactory agreement.

10

Pc=10MPa e x p ~

~, 6 4

2f O_

0,0

simulation

!

!

!

!

0,2

0,4

0,6

0,8

1,0

E1(%) Figure 6a: s~ulafion of undt~ned maxial test - stress deviator

266 ~6

Pc=10MPa ~:4

simul~

2

f

experiment

0-

I

0,0

•

t

0,2

,

l

0,4

,

l

0,6

0,8

1,0 El(%)

Figure 6b: simulation of undrained triaxial test - pore pressure

16

Pc=20MPa

12 "

8

4 .___~ experiment experiment O-

J

0,0

0,5

1,0

1,5 El(%)

Figure 7a: simulation of undrained triaxial test - stress deviatox

10

4 sim7 2 0

. ~ ~

0,0

•

e~f~em •

.

!

0,5

.

.

.

.

!

1,0

.

.

.

.

1,5 E1 (%)

Figure 7b: simulation of undrained tdaxial test - pore pressure

267 INFLUENCE OF PARAMETER PERTURBATION In order to test the stability of the constitutive model with respect to perturbations of its parameters, we have to study the influence of parameters on the model simulation for the experimental tests. We have chosen four parameters which have generally a more significant influence than the others; c, n , m and 13. Because the porosity of chalk has a major influence on mechanical behaviour, it is very interesting to study the relationships between model parameters and porosity. The physical significance of these parameters and their qualitative relations with the porosity are thus mentionned. Parameter

c :

This parameter is introduced in the collapse plastic strain hardening law. It defines the hardening rate of material. Its value decreases when the porosity of rock increases. In figure 8, the simulation of a hydrostatic compression test with three values of c is presented. It can be noticed that the model simulation is stable in spite of the relatively important variation of c. Parameter

n :

This parameter is also introduced in the collapse plastic strain hardening law and represents the plastic collapse rate. It increases when the porosity of chalk increases. In figure 9, we present the simulation of a hydrostatic compression test with three values of n. We notice, once again, that the model simulation is not very sensitive to the variation of n. Parameter m :

This parameter is used in the deviatoric yield function and the failure criterion. It defines the shape of the deviatoric yield surface and the dependence of the failure criterion with hydrostatic stress. Its value increases generally when the porosity of chalk increases. In figure 10, the simulation of a triaxial compression test with a confining pressure of 20MPa by using three values of m is presented and we notice that the model is also relatively stable.

Parameter ~ :

This parameter describes the dependence of the deviatoric strain hardening of the chalk with hydrostatic stress. It increases in general when the porosity of chalk increases. In order to show the influence of the perturbation of this parameter on the model predictions, we simulated a proportional loading test ( k=1.66 ) using three values of 13. The obtained results are presented in figure 11. It can be seen that the volumetric strain is quasi insensitive to 13and the axial strain is also stable with respect to relatively important perturbations of 13.

268

c = 13.3

60

!ol ~

•

c=6.5

[]

c =26.5

0 0

5

10 15 20 V o l u m e t r i c strain (%)

Figure 8: influence of parameter c

60

n = 0.37

.sO

n = 0.27 n = 0.47 30

i°

*.

20

0 0

$

10 15 20 Volumetric strain (%)

Figure 9: influence o f parameter n

25 -4a- m - 0.15

gh

~

20

•"*- m = 0.10

15

-a"0"

..m- m = 0.10 -o- m = 0.20

.~"~'- 105

~

m = 0.20 m = 0.15

0 10

5 0 V o l u m e t r i c strain ( % )

5 10 Axial strain (%)

Figure 10: influence of parameter m

269

Il

20

;

10

="

8

6

n

4

Ev(%)

2

0

2

4

11=0.0008 8---0.00053 B=0.00026 B=0.0008 8--0.00053 8--0.00026

6

E1(%)

Figure 11: influence of parameter 13

VALIDATION OF THE MODEL FOR H O L L O W CYLINDER TEST Hollow cylinder test is chosen to examine the validity of the elastoplastic model for a typical boundary condition problem. In figure 12 the experimentalsystem used for these tests is presented. In this system, we can independently change the interior pressure Pin. the exterior pressure Pex, the axial load Fa and the pore pressure P , . Three kinds of tests with different stress paths have been carried out on the above mentionned white chalks. The dimensions of the samples used in these tests are : exterior diameter De = 110 mm interior diameter Di = 40 mm height H = 120 mm As in the homogeneous tests [ 4 ], the samples were fwsdy dried at 100 °(2 for 24 hours and then saturated under vacuum with methyl alcohol. The constitutive model has been introduced into a finite element program [ 4 ]. Then the simulation of the hollow cylinder tests have been performed using this program in axisyrnmetric condition. In the first test, from a homogeneous stress state (Pin = Pex= 8MPa), Pin and Pex are respectively increased with a constant ratio (APex/APin = k = 8.33 ).The piston of the cell in which the sample is placed is not in contact with the upper platen in this test. The bulk volumetric strain is measured according to the volume of fluid that is expelded from the sample, as a function of the pressures Pin and Pex- In figure 13, comparisons between experimental results and numerical predictions are given. We notice that the model seems to underestimate the initial elastic limit of chalk observed in this test. But as a whole, the accuracy of the model is satisfactory.

270 screw

upper platen

Pex

xterior embrane

Figure 12 : experimental system for hollow cylinder tests 3O

j

20

10 y 0

simulation •

0

!

2

,

I

4

,

I

6

•

I

8

°

I

I

10 12 Ev (%)

Figure 13 : simulation of hollow cylinder test 1 In the second test, the sample is firstly submitted to a non homogeneous stress state (Pex > Pin ; Pex = 20 MPa, Pin = 10 MPa ). Then the contact is made between the piston of the cell and the upper platen, the axial load F a is increased by a hydraulic test machine. The axial displacement is measured as a function of the axial load F a. In figure 14, comparisons between the experiment and the model prediction are presented. It can be seen that the model seems to overestimate the failure stress of chalk as compared to that observed in this test. But a good agreement is also obtained.

globally,

271 In the last test, the exterior pressure Pex is only increased with Pin = 0 and the piston of the cell is not in contact with the upper platen of the sample. Only the tangential strain ~ t on the exterior surface of the sample is measured using strain gauges. The objective of this test is to examine the performance of the model for relatively lower confining pressure problems. In figure 15, comparisons between the experiment and the simulation are presented. There is a very good agreement.

2O e~ g~

10

s

i

~ experiment

_

0

I

i

i

2

4

6

8

Uz (mm) Figure 14: simulation of hollow cylinder test 2

12 10

g.

s

~ O_

0

simulation |

1

I

I

2

=

I

3

=

I

4

=

5

Eft (10E-03) Figure 15: simulation of hollow cylinder test 3

272 CONCLUSION A constitutive model for chalk has been developed using two plastic yield mechanisms; a collapse plastic one to describe the progressive destruction of pores and a deviatoric one for the distorsion of chalk. Proportional loading test, reduced triaxial compression test ,oedometric test and undrained triaxial test have been performed to examine the validity of constitutive model for homogeneous cases. Comparisons between experimental results and numerical responses have shown a general good agreement for these stress paths. The influence of perturbations of the model parameters has been studied. We have noticed that the model predictions are not very sensitive to the relatively important perturbations of the parameters. The validity of the model for a typical boundary condition problem has been tested by using hollow cylinder test. As a whole, good agreements between the model simulation and the experiment have been obtained.

REFERENCES 1.

Shao J.F. , Henry J.P., Development of an elastoplastic model for porous chalk, accepted for publication in the Int. J. of Plasticity_, (1990)

2.

Lade P.V., Elasto-plastic stress strain theory for cohesionless soil with curved yield surfaces, Int. J. Solids Structures. 13, (1977), 1019-1035

3.

Charlez Ph., Rock Mechanics. Theoretical Foundamentals. First proof, (1991)

4.

Shao J.F., Etude du comportement d'une craie blanche tr~s poreuse et mod61isation, Ph.D. Thesis. University of Lille, (1987)

5.

Elliot G. M., Brown E. T., Yield of a soft G6otechniou¢,35. No.4, (1985), 412-423

6.

Jones M.E., Leddra M.J., Compaction and flow of porous rocks at depth, Proc. of the irlternadonal Symposium of Rock Mechanics at great depth . Pau France (1989), 891-898

7.

Nova R. and Hueckel T., A model for soil bahaviour in plastic and hystercfic ranges, Proc. of the international workshoa on Constitutive relations for soils. Grenoble, France (1982), 289-330

8.

Dcsai C.S., Sakami M.R., A constitutive model and associated testing for soft rock, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.. 24, No.5, (1987), 299,307

9.

Lade P.V., Modelling of rock strength in three dimensions, Int. J. Rock Mech. Min. Sei. & Geomeeh. Absm. 21. (1984), 21-33

high porosity rock,

Received 14 May 1990; revised version received 10 September 1990; accepted 15 September 1990